Topological structure of a class of planar self-affine sets

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• 作者：Yuan Zhang ^{yzhangccnu@sina.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Self-affine sets ; Connected component ; Periodic extension
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1524-1536
• 全文大小：1513 K

文摘

Let T=T(A,D^⁎)$T=T(A,{\mathcal{D}}^{⁎})$ be a disk-like Z²${\mathbb{Z}}^2$-tile generated by an expanding 2×2$2\times 2$ matrix A   and a digit set D^⁎⊂Z²${\mathcal{D}}^{⁎}\subset {\mathbb{Z}}^2$. We study the subset F of T   defined by 2303d1b2076aa2085c" title="Click to view the MathML source">AF=F+D$AF=F+\mathcal{D}$, where D⊊D^⁎$\mathcal{D}⊊{\mathcal{D}}^{⁎}$ is a sub-digit set. By studying a periodic extension H=F+Z²$H=F+{\mathbb{Z}}^2$, we classify F into three types according to their topological properties, which generalizes a result of Lau et al. [13]. We also provide some simple criteria for such classification.